Chapter 04

Chapter 04 - Introduction In this chapter, we will...

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FIN 3220A FIN 3220A Actuarial Models I Actuarial Models I Chapter 4 Chapter 4 Benefit Premiums Benefit Premiums FIN 3220A (2005 -2006) 2006) Chapter 4 Benefit Premiums Page Page 2 Introduction Introduction z In this chapter, we will determine the level of life annuity payments necessary to buy, or fund, the benefits of an insurance or annuity contract z By combining the ideas of APV of the payments of various life insurance and annuities z In practice, individual life insurance is usually purchased by a life annuity of contract premiums – the insurance contract specifies the premium to be paid z Contract premiums provide for z Benefits z Expenses of initiating and maintaining the insurance z Margins for profit and for offsetting possible unfavorable experience z We will consider only the premiums for benefits in this chapter FIN 3220A (2005 -2006) 2006) Chapter 4 Benefit Premiums Page 3 Introduction Introduction z A premium principle has to be adopted in the determination of the insurance premium z We will first look at two principles which are based on the impact of the insurance on the wealth of the insurer z The random variable, L , which gives the PV at issue of the insurer’s loss is the key in the model for the principles z Assuming that the insurance is contracted at a certain premium level z Principle I requires that the loss random variable be positive with no more than a specified probability z Principle II is based on the expected utility of the insurer’s wealth FIN 3220A (2005 -2006) 2006) Chapter 4 Benefit Premiums Page Page 4 Introduction Introduction z We usually denote the annual premium as P z Principle I: P will be the least annual premium such that the insurance has probability of a positive financial loss ( L > 0) of at most p z Principle II: P will be the annual premium such that the insurer, using a utility of wealth function u ( x ) = x , will be indifferent between accepting and not accepting the risk
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FIN 3220A (2005 -2006) 2006) Chapter 4 Benefit Premiums Page 5 A Simple Example A Simple Example z An insurer is planning to issue a policy to a life age 0 whose curtate-future-lifetime, K , is governed by the p.f. z The policy will pay 1 unit at the end of the year of death in exchange for the payment of a premium P at the beginning of each year, provided the life survives z Assume that the insurer use an annual effective interest rate of i = 0.06 z For K = k and an arbitrary premium, P , the PV of the financial loss at policy issue is 0 0.2 0,1,2,3,4 k qk == () 1 1 1 1 k k k lk v Pa PP vk dd + + + = ⎛⎞ =+ = ⎜⎟ ⎝⎠ ±± FIN 3220A (2005 -2006) 2006) Chapter 4 Benefit Premiums Page Page 6 A Simple Example A Simple Example z The corresponding loss random variable is z Principle I with p = 0.25: z Since v k +1 is a decreasing function in k , l ( k ) decreases as k increases z The requirement of Principle I will hold if P is such that z Then the financial loss is positive for only K = 0, which has probability 0.2 < 0.25 = p z Thus we have 1 1 K K Lv P a + + =− 2 2 0 vP a −= 2 22 1 0.45796 v P as === FIN 3220A (2005 -2006) 2006) Chapter 4 Benefit Premiums Page 7 A Simple Example A Simple Example z Principle II: z
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This note was uploaded on 05/21/2011 for the course FIN 3220 taught by Professor Cswong during the Spring '05 term at CUHK.

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Chapter 04 - Introduction In this chapter, we will...

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